To evaluate [tex]\(\log_4(1024)\)[/tex] without using a calculator, follow these steps:
1. Understand the Problem Statement:
[tex]\[
\log_4(1024) = x
\][/tex]
This implies that [tex]\(4^x = 1024\)[/tex]. Our goal is to find the value of [tex]\(x\)[/tex].
2. Express 1024 as a Power of 4:
To solve [tex]\(4^x = 1024\)[/tex], it helps to express 1024 as a power of 4. We breakdown 1024 into factors of 4:
[tex]\[
1024 = 4 \times 4 \times 4 \times 4 \times 4 = 4^5
\][/tex]
Therefore, we can write:
[tex]\[
4^5 = 1024
\][/tex]
3. Equate the Exponents:
Since we have [tex]\(4^x = 4^5\)[/tex], we equate the exponents because the bases (4) are equal:
[tex]\[
x = 5
\][/tex]
4. Conclusion:
Therefore, the value of [tex]\(\log_4(1024)\)[/tex] is:
[tex]\[
\log_4(1024) = 5
\][/tex]
Thus, the final result is:
[tex]\[
\log_4(1024) = 5
\][/tex]
